The local Langlands correspondence of essentially unipotent supercuspidal representations for disconnected reductive groups

TL;DR

This paper constructs and generalizes the local Langlands correspondence for essentially unipotent supercuspidal representations, including disconnected reductive groups, under the framework of rigid inner forms.

Contribution

It extends the local Langlands correspondence to disconnected groups and incorporates rigid inner forms, improving upon previous work by including rigidifications of inner twists.

Findings

Constructed the correspondence for disconnected reductive groups.

Proved functoriality and compatibility properties.

Enhanced the framework for explicit local Langlands correspondence.

Abstract

We construct the local Langlands correspondence of essentially unipotent supercuspidal representations under the framework of rigid inner forms and prove a certaion functoriality and compatibilities. This result is stronger than the analogous one in [Sol20], which did not care about rigidifications of inner twists. We also generalize this correspondence for disconnected reductive groups. We expect to use this result for extension of the explicit local Langlands correspondence in [Kal21] for more general supercuspidal representations.